Integral points in orbits in characteristic $p$

Carney, Alexander; Hindes, Wade; Tucker, Thomas J.

doi:10.48550/arxiv.2108.03123

Cited by 2 publications

(5 citation statements)

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“…It is worth pointing out that Theorem 1.3 implies that the forward orbit of any wandering point contains only finitely many polynomials (corresponding to the number of n such that deg(b n ) = 0). This finiteness statement also follows from recent work in [6], though the result above is much stronger. On the other hand, the assumptions in [6] are weaker and the results there apply more generally to function fields K/F p (t) and to more general notions of integral points.…”

Section: − Log |α − R| H(r)supporting

confidence: 70%

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Dynamical Diophantine Approximation Exponents in Characteristic $p$

Hindes¹

2022

Preprint

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Let φ(z) be a non-isotrivial rational function in one-variable with coefficients in Fp(t) and assume that γ ∈ P 1 (Fp(t)) is not a post-critical point for φ. Then we prove that the diophantine approximation exponent of elements of φ −m (γ) are eventually bounded above by ⌈d m /2⌉ + 1. To do this, we mix diophantine techniques in characteristic p with the adelic equidistribution of small points in Berkovich space. As an application, we deduce a form of Silverman's celebrated limit theorem in this setting. Namely, if we take any wandering point a ∈ P 1 (Fp(t)) and write φ n (a) = an/bn for some coprime polynomials an, bn ∈ Fp[t], then we prove thatwhenever 0 and ∞ are both not post-critical points for φ. In characteristic p, the Thue-Siegel-Dyson-Roth theorem is false, and so our proof requires different techniques than those used by Silverman.

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Section: − Log |α − R| H(r)supporting

confidence: 70%

“…This finiteness statement also follows from recent work in [6], though the result above is much stronger. On the other hand, the assumptions in [6] are weaker and the results there apply more generally to function fields K/F p (t) and to more general notions of integral points.…”

Section: − Log |α − R| H(r)supporting

confidence: 70%

Dynamical Diophantine Approximation Exponents in Characteristic $p$

Hindes¹

2022

Preprint

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“…We point out here that an analog of Theorem 4.6 also holds for function fields of curves; see [11,38] for the case of characteristic 0 and p. Here we need to assume that ϕ is not isotrivial. THEOREM 4.7.…”

Section: 4mentioning

confidence: 99%

“…Remarkably though, the reason for the said failure is arithmetic. Our proof of Theorem 1.3 relies on results concerning integral points in orbits, established by Silverman [50,Theorem B] over number fields and in [11,38] for function fields of characteristic zero and p respectively. These allow us to generate infinitely many primitive divisors of certain dynamical sequences, which in turn yield measures corresponding to non-archimedean places with non-trivial potentials.…”

Section: Dynamical Pairs Are Typically Not Adelicmentioning

confidence: 99%

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Quasi-adelic measures and equidistribution on

Mavraki

YE²

2022

Ergod. Th. Dynam. Sys.

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Baker and Rumely, Favre, Rivera, and Letelier, and Chambert-Loir proved an important arithmetic equidistribution theorem for points of small height associated to an adelic measure. To broaden the scope in which arithmetic equidistribution may be employed, we generalize the notion of an adelic measure to that of a quasi-adelic measure and show that arithmetic equidistribution holds for quasi-adelic measures as well. We exhibit examples of non-adelic, quasi-adelic measures arising from the dynamics of quadratic rational maps. In fact, we show that the measures that arise in applications of arithmetic equidistribution theorems are typically not adelic. Finally, we motivate our definition of a quasi-adelic measure by relating it to a seemingly different problem in arithmetic dynamics arising from results of Call, Tate, and Silverman in the study of abelian varieties.

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Integral points in orbits in characteristic $p$

Cited by 2 publications

References 22 publications

Dynamical Diophantine Approximation Exponents in Characteristic $p$

Dynamical Diophantine Approximation Exponents in Characteristic $p$

Quasi-adelic measures and equidistribution on

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